\section{Outstanding Issues}
\label{sec:issues}

Investigation notes synthesised in \texttt{BUG\_FIX\_SUMMARY.md} and \texttt{FINAL\_REPORT.md} isolate two critical blockers to full reproduction. Both issues were explored through targeted experiments and diagnostic scripts, yet neither has been fully resolved.

\subsection{Accuracy Discrepancy in 1D}

The one-dimensional Helmholtz solver consistently returns maximum errors on the order of $10^{0}$ despite matching the paper's configuration (four subdomains, 100 collocation points per subdomain, 100 hidden parameters, $R_m=3.0$, seed 1). Extensive checks confirmed the correctness of automatic differentiation, source-term derivations, operator sign conventions, and equation assembly. Introducing an output bias worsened the error dramatically, implying that the paper's bias-free formulation is not the culprit. The root cause likely lies in an undocumented implementation detail such as the precise random weight initialisation, normalisation strategy, or solver regularisation.

\subsection{Performance Bottleneck in 2D}

The two-dimensional Helmholtz operator leverages nested \texttt{GradientTape} contexts inside a loop over the hidden-layer width. For $M=400$ and 625 collocation points this results in roughly 250{,}000 derivative evaluations per residual assembly, explaining the observed timeout. A vectorised reformulation---computing Jacobians and Hessians for all hidden units in a single batched call---has been designed but not yet committed, pending stabilisation of the codebase after separate experiments.
